If a state can be a superposition of energy states, and mass equals energy (special relativity), and mass curves space-time (general relativity), then could we say that space-time around a quantum system that is in a superposition of states is also in a "superposition of curvatures"?

Good question. Absolutely, save for the fact that as of yet we haven't found or interpreted experimental results about gravity in the quantum realm.
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kηivesJul 20 '12 at 16:57

Somebody did this experimentally--- they did a Cavendish experiment triggered by quantum atomic decay. Of course they didn't see gravity from the other universe. Of course, objective collapse proponents would say that the macroscopic object collapsed.
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Ron MaimonJul 21 '12 at 3:14

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You would ask most physicists (except sir Roger Penrose) and they will tell you that you need planck scale energies to measure quantum gravity

I would dare to suggest instead a gravitational generalization of the schr¨odinger cat and the cavendish experiment mashup:

take in vacuum space, some mass $M$ of the same order as the one used by cavendish do estimate $G$. Now have some quantum system of two states coupled to a system (the tricky experimental part of the setup) that will provide a thrust to $M$ or not depending of the measured eigenvalue of the quantum system

if the thrust system really does not decohere significantly with the environment, you should have the mass (just like the cat) in a superposition of states of different position. So the spacetime curvature must be in a superposition as well

now place test masses nearby. Does the phase in the different eigenstates of $M$ affect the spacetime? well it does affect the electromagnetic field, otherwise we wouldn't see interference terms of light, so it should affect gravity as well.

We should see interference terms in the gravitational field. Not that hard to detect, if you think that Cavendish did this measurement (less the quantum superposition part) in 1797!!!

"so it should affect gravity as well." Careful. The EM equations are linear but the GR equations aren't. I'm still thinking about this. Superposition holds only for linear systems, correct?
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Alfred CentauriJul 20 '12 at 18:31

@AlfredCentauri, the linear field approximation should be entirely valid in this regime, where the gravitational $h_{\mu \nu}$ perturbation is expressed as a Green integral over the retarded sources, just like EM
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diffeomorphismJul 20 '12 at 18:33

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@AlfredCentauri, the nonlinearity of gravity to quantum superposition is just as irrelevant as the nonlinearity of dielectric nonlinear terms to superpositions of light states; in the hamiltonian view, such corrections just shift and transform the spectra, but the eigenstates of those shifted hamiltonians are always in linear superpositions
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lurscherJul 20 '12 at 19:39

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@lurscher, indeed and thanks. While out mowing for awhile, I worked things out.
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Alfred CentauriJul 20 '12 at 22:36